Lemma 11 (Lemma 3.6, [1]}) 

Given a locally Lipschitz definable function \(f\)  with an open domain \(D\in \lbrace x|\Vert x\Vert >1\rbrace \) , for any \(c\) , \(\eta >0\) , there exists \(a>0\)  and a definable desingularizing function \(\Psi \)  on \([0,a)\)  (that is, \(\Psi (x)\in C^1((0,a))\cap C^0([0,a))\)  with \(\Psi (0)=0\) ), such that,
\(\Psi ^{\prime }(f(x))\Vert x\Vert \Vert \bar{\partial } f(x)\Vert  \ge 1, \text{ if } f(x)\in (0,a), \text{ and } \left\Vert \bar{\partial }_{\perp } f(x)\right\Vert  \ge c\Vert x\Vert ^{\eta }\left\Vert \bar{\partial }_{ \backslash \backslash } f(x)\right\Vert .\) 

